Empirical Estimator

Use historical data to estimate assets’ expected return and covariance matrix.

Data

from plotly.io import show
from sklearn.model_selection import train_test_split

from skfolio import Population, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.moments import DenoiseCovariance, ShrunkMu
from skfolio.optimization import MeanRisk, ObjectiveFunction
from skfolio.preprocessing import prices_to_returns
from skfolio.prior import EmpiricalPrior

prices = load_sp500_dataset()
X = prices_to_returns(prices)
X_train, X_test = train_test_split(X, test_size=0.33, shuffle=False)

X_train.head()

	AAPL	AMD	BAC	BBY	CVX	GE	HD	JNJ	JPM	KO	LLY	MRK	MSFT	PEP	PFE	PG	RRC	UNH	WMT	XOM
Date
1990-01-03	0.007576	-0.030303	0.008045	0.118056	-0.016229	-0.001876	0.003581	0.004072	0.033589	-0.014318	0.000000	0.015896	0.005208	-0.009709	0.002938	-0.001813	0.0	-0.019355	0.000000	-0.010079
1990-01-04	0.003759	-0.015500	-0.021355	-0.012422	-0.012831	-0.005639	0.006244	0.002028	0.003991	-0.004993	-0.005557	-0.015647	0.028497	-0.009804	0.016602	-0.019725	0.0	-0.009868	-0.005201	-0.009933
1990-01-05	0.003745	-0.031996	-0.021821	0.000000	-0.014855	-0.009452	-0.013298	-0.010408	0.003975	-0.008212	-0.010874	-0.020641	-0.025189	-0.013991	-0.008646	-0.018004	0.0	-0.043189	-0.010732	-0.005267
1990-01-08	0.003731	0.000000	0.005633	-0.075472	0.009424	0.005725	-0.009883	0.016944	0.000000	0.021159	0.000000	0.012839	0.015504	0.018118	-0.008721	0.018334	0.0	-0.020833	0.013630	0.015381
1990-01-09	-0.007435	0.016527	0.000000	0.000000	-0.007469	-0.020803	-0.026316	-0.031026	-0.031957	-0.007658	-0.011147	-0.007893	-0.002545	-0.013722	-0.021505	0.000000	0.0	-0.024823	-0.026619	-0.020114

Model

Maximize Sharpe Ratio with Empirical Estimator.

max_sharpe_model_empirical = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    prior_estimator=EmpiricalPrior(
        mu_estimator=ShrunkMu(), covariance_estimator=DenoiseCovariance()
    ),
    portfolio_params=dict(name="Empirical - ShrunkMu & DenoiseCovariance")
)
max_sharpe_model_empirical.fit(X_train)
max_sharpe_model_empirical.weights_

array([5.29957573e-02, 5.62258516e-07, 2.19431588e-07, 5.78329525e-02,
       1.05704853e-01, 6.42635887e-07, 1.25145591e-02, 1.64813030e-01,
       3.95272190e-07, 8.40687639e-02, 1.20296728e-06, 1.33374541e-06,
       6.51482298e-02, 7.44911613e-02, 7.02327402e-06, 1.27177410e-01,
       3.87852705e-02, 6.81199403e-02, 4.34872838e-02, 1.04849409e-01])

Benchmark: Maximize Sharpe Ratio with default moments estimators.

benchmark_model = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    portfolio_params=dict(name="Benchmark - Moments Estimators")
)
benchmark_model.fit(X_train)
benchmark_model.weights_

array([9.43631399e-02, 1.13184579e-06, 5.04970598e-07, 1.20834667e-01,
       3.18126275e-02, 8.57806907e-07, 7.11596802e-04, 1.24104939e-01,
       9.49223801e-07, 2.77547553e-02, 1.23409042e-06, 1.37593860e-06,
       1.16299875e-01, 5.73516411e-02, 9.58498590e-06, 1.09493919e-01,
       8.64761638e-02, 1.83992252e-01, 1.32350165e-02, 3.35537683e-02])

Prediction

pred_empirical = max_sharpe_model_empirical.predict(X_test)
pred_benchmark = benchmark_model.predict(X_test)

Analysis

Naive moments estimator outperforms ShrunkMu + DenoiseCovariance.

population = Population([pred_empirical, pred_benchmark])
population.plot_cumulative_returns()